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arXiv:cond-mat/9903258 · Physical Review Letters

Charge-Density-Wave Ordering in Half-Filled High Landau Levels

Edward H. Rezayi, F. D. M. Haldane, Kun Yang

Published Aug 9, 1999• 140 citations
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Abstract

We report on numerical studies of two-dimensional electron systems in the presence of a perpendicular magnetic field, with a high Landau level (index $N\ensuremath{\ge}2$) half filled by electrons. Strong and sharp peaks are found in the wave-vector dependence of both the static density susceptibility and the equal-time density-density correlation function, in finite-size systems with up to twelve electrons. Qualitatively different from the partially filled lowest ( $N\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}0$) Landau level, these results are suggestive of a tendency toward charge-density-wave ordering in these systems. The ordering wave vector is found to decrease with increasing $N$.

DOI
10.1103/physrevlett.83.1219

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We report on numerical studies of two-dimensional electron systems in the presence of a perpendicular magnetic field, with a high Landau level (index N\ensuremath≥2N\ensuremath{\ge}2N\ensuremath≥2) half filled by electrons. Strong and sharp peaks are found in the wave-vector dependence of both the static density susceptibility and the equal-time density-density correlation function, in finite-size systems with up to twelve electrons. Qualitatively different from the partially filled lowest ( N=0N\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}0N=0) Landau level, these results are suggestive of a tendency toward charge-density-wave ordering in these systems. The ordering wave vector is found to decrease with increasing NNN.
The analyticity of the lowest Landau level wave functions and the relation between the filling factor and the total angular momentum severely limits the possible forms of trial wave functions of a disk of electrons subject to a strong perpendicular magnetic field. For the number N of electrons up to 12 we have tested these disk trial wave functions for the half-filled Landau level using Monte Carlo and exact diagonalization methods. The agreement between the results for the occupation numbers and ground state energies obtained from these two methods is excellent. We have also compared the profile of the occupation number near the edge with that obtained from a field-theoretical method. The results give qualitatively identical edge profiles. Experimental consequences are briefly discussed.
We study the effects of disorder on the quantum Hall stripe phases in half-filled high Landau levels using exact numerical diagonalization. We show that, in the presence of weak disorder, a compressible, striped charge-density wave, becomes the true ground state. The projected electron-density profile resembles that of a smectic liquid. With increasing disorder strength W, we find that there exists a critical value, ${W}_{c}\ensuremath{\sim}{0.12e}^{2}/\ensuremath{\epsilon}\mathcal{l},$ where a transition/crossover to an isotropic phase with strong local electron-density fluctuations takes place. The many-body density of states are qualitatively distinguishable in these two phases and help elucidate the nature of the transition.
Finite-size exact diagonalization of the Hamiltonian for four, eight and 16 electrons in a half-filled Landau level under periodic boundary conditions in a rectangular geometry was carried out. The magnitude of the peak value of the structure factor is found to increase with the system size to the power 0.36, consistent with a picture of a quasi-solid discussed by Chui. The energy difference of the lowest-energy state of the two different particle-hole symmetry manifolds decreases as the system size is increased, suggestive of broken particle-hole symmetry.
We consider a two-dimensional strongly correlated electronic system in a strong perpendicular magnetic field at half-filling of the lowest Landau level (LLL). We seek to build a wave function that, by construction, lies entirely in the Hilbert space of the LLL. Quite generally, a wave function of this nature can be built as a linear combination of all possible Slater determinants formed by using the complete set of single-electron states that belong to the LLL. However, due to the vast number of Slater determinant states required to form such basis functions, the expansion is impractical for any but the smallest systems. Thus, in practice, the expansion must be truncated to a small number of Slater determinants. Among many possible LLL Slater determinant states, we note a particular special class of such wave functions in which electrons occupy either only even, or only odd angular momentum states. We focus on such a class of wave functions and obtain analytic expressions for various quantities of interest. Results seem to suggest that these special wave functions, while interesting and physically appealing, are unlikely to be a very good approximation for the exact ground state at half-filling factor. The overall quality of the description can be improved by including other additional LLL Slater determinant states. It is during this process that we identify another special family of suitable LLL Slater determinant states to be used in an enlarged expansion.
The ground-state density of states of the half-filled Falicov-Kimball model contains a charge-density-wave gap. At finite temperature, this gap is not immediately closed, but is rather filled in by subgap states. For a specific combination of parameters, this leads to a stable phase where the system is in an ordered charge-density-wave phase, but there is high density of states at the Fermi level. We show that this property can be, in finite dimensions, traced to a crossing of sharp states resulting from the single particle excitations of the localized subsystem. The analysis of the inverse participation ratio points to a strong localization in the discussed regime. However, the pronounced subgap density of states can still lead to a notable increase of charge transport through a finite size system. We show this by focusing on the transmission in heterostructures where a Falicov-Kimball system is sandwiched between two metallic leads.
We report finite cluster exact diagonalization studies under periodic boundary conditions of two layers of electrons in a strong magnetic field at a half-filled Landau level. For the system studied by Suen et al., the ground states have more than 97 per cent of its spectral weight in the symmetric single particle levels. The eigenfunctions and eigenvalues up to a constant factor for different aspect ratios are essentially the same as previous numerical results of a single layer of electrons of zero thickness at half-filling.